Spectral flow, Brouwer degree and Hill's determinant formula

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Portaluri, Alessandro¹  Wu, Li² 

[1] Univ Torino, DISAFA, Largo Paolo Braccini 2, I-10095 Turin, Italy

[2] Shandong Univ, Dept Math, Jinan 250100, Shandong, Peoples R China

关键词: Brouwer degree;    Trace formula;    Spectral flow;    Hill?s determinant formula;    Elliptic boundary value problems;   

DOI  :  10.1016/j.jde.2020.05.030

来源: Elsevier

【 摘 要 】

In 2005 a new topological invariant defined in terms of the Brouwer degree of a determinant map, was introduced by Musso, Pejsachowicz and the first name author for counting the conjugate points along a semi -Riemannian geodesic. This invariant was defined in terms of a suspension of a complexified family of linear second order Dirichlet boundary value problems. In this paper, starting from this result, we generalize this invariant to a general self-adjoint Morse -Sturm system and we prove a new spectral flow formula. Finally we discuss the relation between this spectral flow formula and the Hill?s determinant formula and we apply this invariant for detecting instability of periodic orbits of a Hamiltonian system. ? 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free